2019/11/04 by Chi-Kwong Li, Li, Chi-Kwong, Yiu‐Tung Poon +1
Computer Science · Mathematics · #15A60 #47A12 #47A30 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1911.01221
openalex publication_date 2019/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Researchers have identified complex matrices A such that a bounded linear operator B acting on a Hilbert space will admit a dilation of the form A ⊗ I whenever the numerical range inclusion relation W(B) ⊆ W(A) holds. Such an operator A and the identity matrix will span a maximal operator system, i.e., every unital positive map from \rm span \I, A, A^*\ to \cal B(\cal H), the algebra of bounded linear operators acting on a Hilbert space \cal H, is completely positive. In this paper, we identify m-tuple of matrices \bf A = (A1, …, Am) such that any m-tuple of operators \bf B = (B1, …, Bm) satisfying the joint numerical range inclusion W(\bf B) ⊆ \rm conv W(\bf A) will have a joint dilation of the form (A1⊗ I, …, Am⊗ I). Consequently, every unital positive map from \rm span \I, A1, A1^*, …, Am, Am^*\ to \cal B(\cal H) is completely positive. New results and techniques are obtained relating to the study of numerical range inclusion, dilation, and maximal operator systems.